Digital Signal Processing Reference
In-Depth Information
Ca
n
Ca
n
Ca
n
x
[
n
]
x
[
n
]
x
[
n
]
C
C
C
n
n
n
C
C
C
1
a
0
a
1
a
1
Figure 9.19
Discrete-time exponential signals.
x[n] = (-2)
n
.
Figure 9.19 gives the three cases for
a 6 0.
Consider, for example,
Begin-
ning with
n = 0,
the number sequence for
x
[
n
] is
1, -2, 4, -8, 16, -32, Á .
Hence, the number
sequence is exponential with alternating sign. Letting
a =-a
with
a
positive, we obtain
x[n] = Ca
n
= C(-a)
n
= C(-1)
n
a
n
= (-1)
n
x
a
[n],
(9.51)
where denotes the exponential signals plotted in Figure 9.18. Hence, the signals of
Figure 9.19 have the same magnitudes as those of Figure 9.18, but with alternating signs.
x
a
[n]
CASE 2
C
Complex,
a
Complex, with Unity Magnitude
Next we consider the case that
C
and
a
are complex, with
C = Ae
jf
= A∠f,
a = e
jÆ
0
,
(9.52)
where and are real and constant. As defined in Section 9.3, is the normalized
discrete-frequency variable. The complex exponential signal
x
[
n
] can be expressed as
A, f,
Æ
0
Æ
x[n]
= Ae
jf
e
jÆ
0
n
= Ae
j(Æ
0
n+f)
= Acos(Æ
0
n + f) + jA sin(Æ
0
n + f),
(9.53)
from Euler's relation in Appendix D. Recall from Section 9.3 that the sinusoids in (9.53) are
periodic only for with
k
and
N
integers. [See (9.39).] A plot of the real part of
(9.53) is given in Figure 9.20, for f = 0.
Æ
0
= 2pk/N,
A
cos
0
n
A
n
Figure 9.20
Undamped discrete-time
sinusoidal signal.
A
